Optimal. Leaf size=59 \[ \frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 x+2}-\frac {11 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}} \]
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Rubi [A] time = 0.01, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ \frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 x+2}-\frac {11 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 204
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x}+\frac {11}{2} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x}+11 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x}-\frac {11 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{\sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 59, normalized size = 1.00 \[ \frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 x+2}-\frac {11 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 71, normalized size = 1.20 \[ -\frac {11 \, \sqrt {7} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.33, size = 197, normalized size = 3.34 \[ \frac {11}{140} \, \sqrt {5} {\left (\sqrt {70} \sqrt {2} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {280 \, \sqrt {2} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{{\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 108, normalized size = 1.83 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (33 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+22 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+14 \sqrt {-10 x^{2}-x +3}\right )}{14 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 49, normalized size = 0.83 \[ \frac {11}{14} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {\sqrt {-10 \, x^{2} - x + 3}}{3 \, x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.80, size = 716, normalized size = 12.14 \[ \frac {2\,{\left (\sqrt {1-2\,x}-1\right )}^3}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3\,\left (\frac {14\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^4}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {6\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^3}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {12\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{25\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {4}{25}\right )}-\frac {11\,\sqrt {7}\,\mathrm {atan}\left (\frac {2904\,\sqrt {3}\,\sqrt {7}}{875\,\left (\frac {2904\,{\left (\sqrt {1-2\,x}-1\right )}^2}{35\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {53724\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {5808}{175}\right )}+\frac {1452\,\sqrt {7}\,\left (\sqrt {1-2\,x}-1\right )}{875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )\,\left (\frac {2904\,{\left (\sqrt {1-2\,x}-1\right )}^2}{35\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {53724\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {5808}{175}\right )}-\frac {1452\,\sqrt {3}\,\sqrt {7}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{175\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2\,\left (\frac {2904\,{\left (\sqrt {1-2\,x}-1\right )}^2}{35\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {53724\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {5808}{175}\right )}\right )}{7}-\frac {4\,\left (\sqrt {1-2\,x}-1\right )}{5\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )\,\left (\frac {14\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^4}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {6\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^3}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {12\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{25\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {4}{25}\right )}+\frac {37\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2\,\left (\frac {14\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^4}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {6\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^3}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {12\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{25\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {4}{25}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1 - 2 x}}{\left (3 x + 2\right )^{2} \sqrt {5 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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